85jct_catalan

260 E.tiRLINC;tR AND HOFBAL!EH
(5.1) is just (3.5) (5.2) is (3. I ). and (5.3)(5.5) are recurrence relations
extending (1.2).
(3) Surprisingly the Carlitz q-Catalan numbers also are covered by
them, as we recognize from (5.2) by setting u = q ‘, h = q’, .Y = q I. and
comparing with (2.5):
C,,(q
‘; q ?, q2) = q -W,(q)
or
c?(q) = C,(q; q2, q~-2L
But then (5.1) implies that the two statistics “(‘;) - inv” and
“2ct - 2p + des” have the same distribution on %,;,. One can show that they
are also equally distributed on S(n, n). We will give a “bijective proof’ of
this fact in the next section,
(4) It would be interesting to extend the above theorem further by
including the inversions in (5.1).
(5) We suppose that most of the things that are known for the
statistics (des, maj) can be extended to (des, ~1, fi). E.g., it is easy to
generalize (4.6) to
which is the answer to a general Simon Newcomb problem [2,6, 191 for
two different letters.
6. POLYGONS AND THE POLYA-GESSEL q-CATALAN NUMBERS
Until now we have formulated the combinatorics of our q-Catalan numbers
only in terms of O-l-words and lattice paths. In this last section we
describe a further combinatorial model, which is the basis for a q-analog of
the Catalan numbers introduced by Polya [22] and Gessel [lo].
Following Gessel [lo], we consider pairs of lattice paths in the plane,
each path starting at the origin, but now consisting of unit horizontal and
vertical steps in the positive direction.
Let grin., be the set of such path-pairs (TC, (T) with the following properties:
(i) both ‘II and 0 end at the point (j, n -j),
(ii) n begins with a unit vertical step and cr with a horizontal,
(iii) n and u do not meet between the origin and their common
endpoint.